Ngô Quốc Anh

December 31, 2011

A Hardy-Moser-Trudinger inequality: A conjecture by Wang and Ye

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 21:48

Let B denote the standard unit disk in \mathbb R^2. The famous Moser–Trudinger inequality says that

\displaystyle\int_B {\exp \left( {\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}}} \right)dx} \leqslant C < \infty ,\quad\forall u \in H_0^1(B)\backslash \{ 0\}

holds. There is another important inequality in analysis, the Hardy inequality which claims that

\displaystyle H(u) = \int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} \geqslant 0,\quad\forall u \in H_0^1(B)

holds. The one H is usuall called the Hardy functional. One can immediately see that

\displaystyle\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}} \leqslant \dfrac{{4\pi {u^2}}}{{\displaystyle\int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} }}

for any u \in H_0^1(B)\backslash \{ 0\}. Recently, in a paper accepted in Advances in Mathematics journal, Wang and Ye proved that there exists a constant C_0 >0 such that the following

\displaystyle\int_B {\frac{{4\pi {u^2}}}{{H(u)}}dx} \leqslant C_0 < \infty ,\quad\forall u \in \mathcal H(B^n)\backslash \{ 0\}

where B^n is the unit ball in \mathbb R^n, n \geqslant 2 and \mathcal H=\mathcal H(B^n) is the complement of C_0^\infty(B^n) with respect to the following norm \|u\|_{\mathcal H}=\sqrt{H(u)}.

Let us go back to the case n=2. They then defined

\displaystyle {H_d}(u) = \int_\Omega {|\nabla u{|^2}dx} - \frac{1}{4}\int_\Omega {\frac{{{u^2}}}{{d{{(x,\partial \Omega )}^2}}}dx} > 0,\quad \forall u \in H_0^1(\Omega )\backslash \{ 0\}

where \Omega is a regular, bounded and convex domain sitting in \mathbb R^2. They then conjectured that the following

\displaystyle\int_\Omega {\frac{{4\pi {u^2}}}{{{H_d}(u)}}dx} \leqslant C(\Omega ) < \infty ,\quad\forall u \in {\mathcal H_d}(\Omega )\backslash \{ 0\}

still holds for some constant C(\Omega)>0 where {\mathcal H_d}(\Omega ) denotes the completion of C_0^\infty (\Omega) with the corresponding norm associated with H_d. Apparently, the conjecture holds true for \Omega = B.

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August 16, 2010

The Moser-Trudinger inequality for domains with holes

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 2:42

In this entry, we are interested in the following result

Theorem (Moser-Trudinger’s inequality for domains with holes). Let \Omega be a bounded smooth domain in \mathbb R^2. Let S_1 and S_2 be two subsets of \overline \Omega satisfying

{\rm dist}(S_1,S_2) \geqslant \delta_0>0

and let \gamma_0 be a number satisfying \gamma_0 \in \left(0,\frac{1}{2}\right). Then for any \varepsilon>0, there exists a constant c=c(\varepsilon, \delta_0, \gamma_0)>0 such that

\displaystyle\int_\Omega {{e^u}} \leqslant C\exp \left[ {\frac{1}{{32\pi - \varepsilon }}\int_\Omega {{{\left| {\nabla u} \right|}^2}} + C} \right]

holds for all u \in H_0^1(\Omega) satisfying

\displaystyle\frac{{\int_{{S_1}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}, \quad \frac{{\int_{{S_2}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}.

(more…)

August 13, 2010

An application of the (Moser-)Trudinger inequality to the mean field equations


Let (M,g) be a compact Riemannian surface with the volume |M|. The simplest form of the mean field equation studied in the contexts of the prescribing Gaussian curvature, statistical mechanics of many vortex points in the perfect fluid and self-dual gauss theories is given by

\displaystyle - {\Delta _g}u = \lambda \left( {\frac{{{e^u}}}{{\int_M {{e^u}d{v_g}} }} - \frac{1}{{|M|}}} \right), \quad \text{ on } M

with

\displaystyle\int_M {ud{v_g}} = 0

where \lambda is a real number.

The mean field equation has a variational structure, and u is a solution if and only if it is a critical point of

\displaystyle {J_\lambda }(v) = \frac{1}{2}\int_M {{{\left| {\nabla v} \right|}^2}d{v_g}} - \lambda \log \int_M {{e^v}d{v_g}}

defined for v \in H^1(M) with

\displaystyle\int_M {vd{v_g}} = 0.

It is worth noticing from this entry that so far our Moser-Trudinger’s inequality is just for \mathbb S^2

Theorem (Moser-Trudinger’s inequality for \mathbb S^2). There are constants \eta>0 and c=c(g)>0 such that for each p \geqslant 2

\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}}  \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left|  {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}

for all u \in W^{1,2}(\mathbb S^2).

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July 8, 2010

The Moser-Trudinger inequality


Followed by an entry where the Trudinger inequality had been discussed we now consider an important variant of it known as the Moser-Trudinger inequality.

Let us remind the Trudinger inequality

Theorem (Trudinger). Let \Omega \subset \mathbb R^n be a bounded domain and u \in W_0^{1, n}(\Omega) with

\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1.

Then there exist universal constants \beta>0, C_1>0 such that

\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega |.

The Trudinger inequality has lots of application. For application to the prescribed Gauss curvature equation, one requires a particular value for the best constant \beta_0. In connection with his work on the Gauss curvature equation, J. Moser [here] sharpended the above result of Trungdier as follows

Theorem (Moser). Let \Omega \subset \mathbb R^n be a bounded domain and u \in W_0^{1, n}(\Omega) with

\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1.

Then there exist sharp constants \beta_0=\beta(n)>0, C_1=C_1(n)>0 given by

\displaystyle \beta_0=n\omega_{n-1}^{\frac{1}{n}-1}

such that

\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega|, \quad \forall \beta \leqslant \beta_0.

The constant \beta_0 is sharp in the sense that for all \beta>\beta_0 there is a sequence of functions u_k \in W_0^{1,n}(\Omega) satisfying

\displaystyle\int_\Omega {{{\left| {\nabla u_k} \right|}^n}dx} \leqslant 1

but the integral

\displaystyle\int_\Omega {\exp (\beta {|u_k|^\frac{n}{n-1}})dx}

grow without bound.

For general compact closed manifold (M,g) the constant on the right hand side of the Moser-Trudinger inequality depends on the metric g. Working on a sphere (\mathbb S^2,g_c) with a canonical metric allows us to control the constants.

Theorem (Moser). There is a universal constant C_1>0 such that for all u \in W^{1,2}(\mathbb S^2) with

\displaystyle\int_{\mathbb S^2} {{{\left| {\nabla u} \right|}^n}dv_{g_c}}  \leqslant 1

and

\displaystyle\int_{\mathbb S^2} u dv_{g_c}=0

we have

\displaystyle\int_{\mathbb S^2} \exp(4\pi u^2)  \leqslant {C_1}.

Observe that

\displaystyle 4\pi = \int_{{\mathbb{S}^2}} {d{v_{{g_c}}}} <\int_{{\mathbb{S}^2}} {{e^{4\pi {u^2}}}d{v_{{g_c}}}} \leqslant {C_1}.

In the same way as we introduce in the entry concerning the Trudinger inequality one can show

Corollary. For

\displaystyle C_2 := \log C_1 +\log\frac{1}{4\pi}

one has

\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}

for all u \in W^{1,2}(\mathbb S^2).

Obviously, C_2 >0 since C_1 >4\pi. It turns out to determine the best constant C_2. This had been done by Onofri known as the Onofri inequality [here].

Theorem (Onofri).Let u \in W^{1,2}(\mathbb S^2) then we have

\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u

with the equality iff

\Delta u +e^{2u}=1.

The proof of the Onofri inequality relies on a result due to Aubin

Theorem (Aubin). For all \varepsilon>0 there exists a constant C_\varepsilon such that

\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\left( {\frac{1}{2} + \varepsilon } \right)\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_\varepsilon }

for any u belonging to the following class

\displaystyle S = \left\{ {u \in {W^{1,2}}({\mathbb{S}^2}):\int_{{\mathbb{S}^2}} {{e^{2u}}{x_j}d{v_{{g_c}}}} = 0,j = \overline {1,3} } \right\}.

Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.

July 1, 2010

The Trudinger inequality

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 10:53

In 1967, Neil S. Trudinger announced a result in J. Math. Mech. (now known as Indiana Univ. Math. J.) which can be seen as a limiting case of the Sobolev inequality [here] or [here].

It is well-known from the Sobolev embedding theorem that

W_0^{\alpha, q}(\Omega) \hookrightarrow L^p(\Omega)

for

\displaystyle \frac{1}{p}=\frac{1}{q}-\frac{\alpha}{n}, \quad q\alpha<n.

The case q\alpha=n is commonly referred to the limitting case. If \alpha=1, n=2 and q<2 we obtain

W_0^{1, q}(\Omega) \hookrightarrow L^p(\Omega).

In general one cannot take the limits q \to 2 and p \to \infty, i.e.

W_0^{1, 2}(\Omega) \not\hookrightarrow L^\infty(\Omega).

A counter-example is given by

\displaystyle \log\left(1+\log\frac{1}{|x|}\right)

(more…)

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